Questions from General Calculus

Q: Determine whether or not the given set is (a) open

Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected. {(x, y) | (x, y) ≠ (2, 3)}

Q: Let F (x, y) = -y i +

Let F (x, y) = -y i + x j/ x2 + y2. (a). Show that ∂P/∂y = ∂Q/∂x. (b). Show that ∫C F ∙ dr is not independent of path. [Hint: Compute ∫C1 F ∙ dr and ∫C2 F ∙ dr, where C1 and C2 are the upper and lowe...

Q: Verify that the Divergence Theorem is true for the vector field F

Verify that the Divergence Theorem is true for the vector field F on the region E. F (x, y, z) = y2z3 i + 2yz j + 4z2 k, E is the solid enclosed by the paraboloid z = x2 + y2 and the plane z = 9

Q: (a). Write the definition of the surface integral of a

(a). Write the definition of the surface integral of a scalar function f over a surface S. (b). How do you evaluate such an integral if S is a parametric surface given by a vector function r (u, v)? (...

Q: Use a computer to graph the parametric surface. Get a printout

Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant.

Q: Use Stokes’ Theorem to evaluate ∫C F ∙ dr.

Use Stokes’ Theorem to evaluate ∫C F ∙ dr. In each case C is oriented counterclockwise as viewed from above. F (x, y, z) = xy i + yz j + zx k, C is the boundary of the part of the paraboloid z = 1 -...

Q: (a). Use Stokes’ Theorem to evaluate ∫C F ∙

(a). Use Stokes’ Theorem to evaluate ∫C F ∙ dr, where F (x, y, z) = x2z i + xy2 j + z2 k and C is the curve of intersection of the plane x + y + z = 1 and the cylinder x2 + y2 = 9, oriented countercl...

Q: (a). Use Stokes’ Theorem to evaluate ∫C F ∙

(a). Use Stokes’ Theorem to evaluate ∫C F ∙ dr, where F (x, y, z) = x2y i + 1/3 x3 j + xy k and C is the curve of intersection of the hyperbolic paraboloid z = y2 - x2 and the cylinder x2 + y2 = 1, o...

Q: Solve the initial-value problem. y'' - 6y' +

Solve the initial-value problem. y'' - 6y' + 10y = 0, y (0) = 2, y' (0) = 3

Q: Solve the initial-value problem. 4y'' - 20y' +

Solve the initial-value problem. 4y'' - 20y' + 25y = 0, y (0) = 2, y' (0) = -3